Geometric Methods in Representation Theory

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

The principal investigator will work on five mathematics research projects in the general area of `Lie Theory and Geometry.' The first project concerns a solution of the celebrated Valiant's conjecture in Geometric Complexity Theory asserting that if the permanent of a square matrix of size n can be realized as a linear projection of the determinant of a square matrix of smallest size m, then the function m grows faster than any polynomial in n. The second project is aimed at solving a conjecture of Hitchin on the nonvanishing of primitive adjoint invariant forms restricted to certain subspaces of a semisimple Lie algebra. This has interesting implications on the geometry of the moduli spaces of semistable principal bundles on smooth projective curves. The third project (jointly with E. Feigin from Russia and P. Littelmann from Germany) aims at studying the graded character of any finite-dimensional irreducible representation (more generally of a Demazure module) of a semisimple Lie algebra. The aim of the fourth project is to study the saturated tensor cone for symmetrizable Kac-Moody Lie algebras. This project is an extension of an earlier project for the semisimple Lie algebras completed by the PI jointly with P. Belkale. The fifth project concerns the completion of a graduate textbook on ``Verlinde formula, its complete proof and consequences." The proof of the Verlinde formula (which was initially conjecturally given by the mathematical physicist E. Verlinde) as well as various applications are scattered through the literature and there is no single source containing these; such a book fill a void in the existing literature. This mathematics research project brings together several areas of mathematics including Topology, Combinatorics, Algebraic Geometry and Representation Theory, and is expected to have significant consequences in Mathematics and in Computer Science. More specifically, the first project on the Valiant's conjecture in Geometric Complexity Theory has significant implications towards a major unsolved problem in Theoretical Computer Science commonly referred to as Cook's ``P versus NP" problem. Informally, it asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer; it is considered to be one the most important open problems in the field. In addition, it is expected that the solution will spawn a lot of activity in the area. Two books authored by the principal investigator (one of these coauthored with M. Brion from France) have become standard texts on the subject. The PI is working towards the completion of a new graduate textbook on ``Verlinde formula, its complete proof and consequences." This would be the very first book on the subject; it is expected that it will serve as a basic source for graduate students and professional mathematicians and mathematical physicists alike thus substantially promoting teaching and learning. The PI has successfully supervised six Ph.D. students and one Master's student; he is currently supervising two Ph.D. students. He has coorganized several prestigious international conferences in Germany and USA. In addition, he is coorganizing an interdisciplinary summer school in Utah aimed mainly at Ph.D. students and postdoctoral fellows in Mathematics and Computer Science.